Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
19 views

How to show contradiction in the Hardy inequality when the singularity power is greater that 2.

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
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0answers
11 views

Prove that if the second x derivative of a homogeneous function of deg 2 exists at the origin, then f(1,0)=f(-1,0)

Let $$f: \mathbb{R} ^2 β†’ \mathbb{R}$$ be a real-valued function on $ \mathbb{R} ^2$ that is homogeneous of degree two. Suppose that f is continuously differentiable on $\mathbb{R} ^2$\ $ {0}$ so that ...
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2answers
46 views

Interior and boundary of $\mathbb{Q}$ in $\mathbb{R}$

The closure of $\mathbb{Q}$ is $\mathbb{R}$, but what are the boundary and interior of $\mathbb{Q}$? I think both are $\mathbb{R}$ because in any open ball centered at $q\in \mathbb{Q}$, there is any ...
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1answer
43 views

Is the function unbounded on $[a,b]$ if there is a removable discontinuity in between $a$ and $b$? [closed]

Since we know that discontinuous function eg. $f(x)=\frac{1}{x}$ is unbounded on $[-1,1]$ because an $M\in\mathbb{R}$ satisfying $|f(x)|\leq M$ doesn't exist for $x=0$. So, I believe the function is ...
0
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1answer
31 views

Does $\lim_{x \to 1^-} \sum_{n=0}^\infty x^{n!} = \infty$?

Does $\lim_{r \to 1^-} \sum_{n=0}^\infty r^{n!} = \infty$? I am working on a complex analysis question that asks to show $\sum_{n=0}^\infty z^{n!}$ cannot be extended past the open unit disk. My ...
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0answers
21 views

Non-convex open set whose closure is convex

In a topological vector space, can there be a non-convex open set whose closure is convex?
-1
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1answer
33 views

If $J$ is an interval then $f(J)$ is an interval (if $f$ is continuous)

Stumbled upon a proof of one lemma. The statement of the lemma is quite obvious, but something in the proof does not seem to make sense very easily for me. Hence my asking. Statement: Let $f$ be ...
0
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2answers
45 views

How can we show the inequality?

Let $h_1,h_2:[0,1]\rightarrow \mathbb{R}$ be continuous functions. How can we show that $h_1(x)\leq h_2(x)$ for each $x\in [0,1]$, given that $h_1(x) \leq h_2(x)$ for each $x\in \mathbb{Q}\cap ...
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0answers
36 views

Showing $y''' + y'^2 = 0$ with $y(0) = 0$, $y'(0) = 1$, $y''(0) = 0$ is asymptotic to $x$ rigorously

Considering the ODE problem given by $$y''' + y'^2 = 0$$ with $y(0) = 0$ and $y'(0) = 1$ and $y''(0) = 0$. Using Mathematica, it seems that numerically, the solution satisfies $y(x) \sim x$ as $x ...
4
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2answers
48 views

Arzela-Ascoli for $\mathbb R^n$ from the case of $\mathbb R$?

In class, we proved the Arzela-Ascoli theorem for $\mathbb R$. The lecturer said it's also true for $\mathbb R^n$, and this version is deducible from $\mathbb R$. I tried to do this but failed. How ...
2
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1answer
47 views

Double sequence, if $(x_m)_m$ and $(y_n)_n$ converge, then they have the same limit?

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, ...
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2answers
82 views

Find all differentiable functions for which $f'(x)+\int_{\pi/4}^x f(t)dt = 0 $

I am having trouble knowing when I have found all possible functions $f(x)$ for the equation. How can I be sure I have found every single one? The question is: Find all differentiable functions ...
0
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1answer
13 views

Getting stuck on an Analysis Question - Limit Theory

The question is: Prove that if $\lim\limits_{n\to\infty}X_n = a$, and $X_n > 0$, $n$ is any natural number, then $\lim\limits_{n\to\infty} \sqrt[n]{X_1\cdot X_2\cdot \cdot \cdot X_n} = a$. I ...
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1answer
304 views

How to prove this series about Fibonacci number?

How to prove this series: I have no idea where to start. $$\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$$ where $F_{1}=1,~F_{2}=1,~F_n=F_{n-1}+F_{n-2},~~n\geq 3$.
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0answers
44 views

Limits of two fixed points of $E_\mu(x) = \mu e^x$

Please let me know if this proof is OK. Problem statement: Given that $E_\mu(x) = \mu e^x$, where $0 < \mu < 1/e$, show that if $q_\mu < p_\mu$ are fixed points, where $q_\mu$ is attractive ...
0
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2answers
39 views

Derivative of function involving absolute value

Could anyone help me with differentiating $|x|^5$ and $\frac{|x|^3}{(1+x^2)^8}$? I used the way we differentiate $|x|$ via substitution, i.e. enter link description here It fails on the two ...
2
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2answers
46 views

Theorem 2.41 in Baby Rudin: Is this proof good enough? Can we generalise it?

Here is Theorem 2.41 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If a set $E$ in $\mathbb{R}^k$ has one of the following three properties, then it has the other ...
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4answers
128 views

Can you find the maximum or minimum of an equation without calculus? [on hold]

Without using calculus is it possible to find provably and exactly the maximum value or the minimum value of a quadratic equation $$ y:=ax^2+bx+c $$ (and also without completing the square)? I'd ...
1
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0answers
52 views

Does $f:\mathbb{R}^d\to \mathbb{C}$ implies $|f|<\infty$ almost everywhere?

I was reading notes on measure theory, and just want some clarifications. If $f:\mathbb{R}^d\to [0,+\infty]$, is it allowed to have $f(x)=+\infty$? What does it mean? Because when I learned the ...
1
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1answer
29 views

which of the following is NOT a possible value of $(e^{f})''(0)$??

Let $f$ be an analytic function on $\bar{𝐷} = \{z \in \mathbb{C}: |z| \le 1\}$. Assume that $|𝑓(𝑧)| ≀ 1$ for each $z\in \bar{D}$. Then, which of the following is NOT a possible value of ...
5
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1answer
65 views

What is so special about the Schwarz Inequality?

I am studying Spivak's Calculus and the first two problem sets have rather lengthy,but very interesting, work-throughs of three proofs for the Schwarz Inequality: $$\sum_{i=1}^{n} x_iy_i ...
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0answers
24 views

Spherical Bessel expansion of Green function

Any reference/advice would be good. I can use eigenfunction to solve the Green function for $$\Delta u(x) + k^2 u(x) = \delta(x - y)$$ boundary condition given as $u = 0$ on $\partial B(1)$, unit ...
3
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4answers
60 views

Textbook for Vector Calculus

Can anyone recommend a textbook for studying vector calculus (vector analysis) only, that focuses on the theoretical mathematics behind vector calculus? Currently, I am using vector analysis by ...
3
votes
2answers
58 views

What is the minimal correction to the harmonic series such that it converges?

as you all hopefully know, the series $$ \sum_{k\ge 1}\frac{1}{k} $$ diverges. Now I know that you can add some logarithmic corrections, such that it converges: $$ \sum_{k\ge 1}\frac{1}{k\log(k)^2} $$ ...
0
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0answers
46 views

Continuous non-differentable functions

I'm looking for some examples of everywhere continuous functions which are nowhere differentable. I found already Takagi curve and Weierstrass function. Can you point out some online courses or pdf ...
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0answers
26 views

Meanvalue theorem for quadratic arguments

I have trouble proving the following result and I would be happy about any kind of suggestion how the precise argument looks like. Let $f : [0,\infty) \rightarrow \mathbb{R}$ denote a continuously ...
1
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1answer
26 views

CDF for Laplace distribution

According to the Wiki article on the Laplace distribution, $$F(x)=\int\limits_{-\infty}^x f(u)du=\begin{cases} \frac{1}{2}\exp(\frac{x-\mu}{b}) && \text{if }x< \mu \\ ...
1
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1answer
16 views

Prove the following simple exponentiation equality.

Having trouble with the following proof. Given $b > 1, c > 0$, prove that $ \exists \; x$ s.t. $b^{x} < c$. We can't use $log$, and I have already shown that $b^{x} > c$ by using the ...
2
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1answer
45 views

Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...
0
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1answer
171 views

Equivalence of dual space of normed space X and continuously differentiable functions.

Define that two normed spaces $X$ and $Y$ are equivalent if there exists bounded linear maps $A: X \to Y$ and $B: Y \to X$ such that $A$ and $B$ are inverses of each other. How do you show that there ...
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0answers
22 views

Convolution of two $L^\infty$ function with compact support.

I have the following lemma without proof: Lemma. Let $f, g \in L^\infty(\mathbb R^n)$ with compact supports. Then $f \ast g \in C(\mathbb R^n)$. Is this even true? I get this: $$ \begin{align*} ...
2
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1answer
38 views

Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
2
votes
1answer
57 views

Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
1
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2answers
25 views

Why is it that $\inf_{x∈E} d(x, a) > 0$?

Suppose $X$ is a metric space, $E βŠ‚ X$ is closed, and $a$ is a point not in $E$. Why is it that $$\inf_{x∈E} d(x, a) > 0$$ ?
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2answers
60 views

How can I show that $(I_1 × · · · × I_n) \setminus \{a\}$ is open and connected for any $a ∈ \mathbb{R}^n$ (with $n \geq 2$)?

Let $I_1, . . . , I_n$ be any open intervals in $\mathbb{R}$ for $n β‰₯ 2$. How can I show that $(I_1 Γ— Β· Β· Β· Γ— I_n) \setminus \{a\}$ is open and connected for any $a ∈ \mathbb{R}^n$?
1
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1answer
38 views

Number of roots of a sequence of a uniformly convergent holomorphic functions implies an upper bound for the number of roots of their limit

Let $G$ be an open, simply connected region in $\mathbb{C}$. We define a sequence of holomorphic functions $(f_n)_{n \in \mathbb{N}}, f_n: G \to \mathbb{C}$ as almost uniformly convergent iff $(f_n)$ ...
0
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2answers
32 views

If $U βŠ‚ \mathbb{R}^n$ is open and $B βŠ‚ U$, then why is it that $B$ relatively open in $U$ if and only if $B$ is open?

If $U βŠ‚ \mathbb{R}^n$ is open and $B βŠ‚ U$, then why is it that $B$ relatively open in $U$ if and only if $B$ is open? I can prove it for closed sets. Does it follow directly from what's below that ...
2
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1answer
35 views

Show that if we had a complete metric space $X$ with no isolated points, then every singleton $\{x\}$ is nowhere dense

My attempt: The closure of the singleton is again the singleton itself Since there are no isolated points, then clearly $\{ x \}$ does not contain any non-empty open set hence the interior of the ...
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0answers
20 views

If $E βŠ‚ \mathbb{R}^n$ and $F βŠ‚ \mathbb{R}^m$ are both connected, why is $E × F$ connected in $\mathbb{R}^{n+m}$?

Question: Suppose that $E βŠ‚ \mathbb{R}^n$ and $F βŠ‚ \mathbb{R}^m$ are both connected. How can I show that $E Γ— F$ is connected in $\mathbb{R}^{n+m}$. My Progress: Let $U βŠ‚ \mathbb{R}^n$ be open with ...
0
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1answer
26 views

Why is this $U_{a_1},…,a_m$ open in $\mathbb{R}_{nβˆ’m}$ for any choice of ${a_1, . . . , a_m}$?

Let $U βŠ‚ \mathbb{R}^n$ be open with respect to the usual topology. Fix any real numbers $a_1, . . . , a_m$ for $m < n$ and consider the set $U_{a_1,...,a_m} := ${ $(x_1, . . . , x_{nβˆ’m}) ∈ ...
1
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1answer
40 views

Analysis question regarding properties of the Riemann integral.

Consider $f:[0,1]\mapsto\mathbb{R}$ such that $|f|^2 \in R([0,1])\cap C^2([0,1])$ and $f(0) = 0$, $f(1) = 0$. Then prove that $$ \left( \int_{0}^{1}|f|^2 dt \right)^\frac{1}{2} \leq 2 \left( ...
2
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1answer
42 views

Square Root of the shift operator indexed by $\mathbb{Z}$

My question is very similar to this question, but instead of indexing by $\mathbb N$ I am indexing by $\mathbb Z$. Consider the shift operator $T : \ell^1(\mathbb Z) \to \ell^1(\mathbb Z) $ given by ...
3
votes
1answer
19 views

Convergence radius and two-times-differentiability of power series.

I wanted to compute the radius of convergence for the following the power series $$\sum_{n=1}^{\infty} a_nz^n$$ with $(i) \, a_n = n!, \, (ii) \, a_n = \sqrt[\leftroot{-3}\uproot{3}n]{n}$ Then I ...
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0answers
31 views

Looking for an (outside $\Bbb R$) application of a certain theorem

I have the following theorem in the lecture notes: Let $E$ be a normed vector space and $\Omega \subset E$ be open and connected, and let $F$ be a Banach space. Let $(f_n)$ be a sequence of ...
1
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2answers
41 views

Simple Inequality of Complex Numbers, $\left| \frac{a-b}{1-\overline{a}b} \right| <1$

Exercise from Ahlfor's Complex: Given $a,b \in \mathbb C$, with $|a| <1$, $|b|<1$, prove: $$\left|\frac{a-b}{1-\overline{a}b}\right| <1.$$ My argument: Lemma: If $\alpha, \beta \in ...
7
votes
2answers
133 views

When is it possible to have $f(x+y)=f(x)+f(y)+g(xy)$?

According to the question mentioned here, it seems that there is no function $f(x)$ such that the functional equation $$f(x+y)=f(x)+f(y)-(xy-1)^2$$ can hold. Motivated by this question, I found it ...
0
votes
1answer
74 views

$f(x,y)=(x^2-y^2,2xy)$ What is $f(A)=B$?

Let $f(x,y)=(x^2-y^2,2xy)$. I have shown that $f$ is one-to-one on $A$ where $A$ is the set consisting of all $(\vec{x},\vec{y})$ such that $x>0$. I now have to find the set $f(A)=B$. I'm not ...
2
votes
1answer
19 views

How to optimize this function here?

How could I minimize the following? $\vec{x} \in \mathbb{R}^8$ $$f(\vec{x}) = 3\frac{|x_1 - x_2||x_3 - x_4| + |x_5 - x_6||x_7 - x_8|}{|(x_1 - x_2)(x_3 - x_4) - (x_5 - x_6)(x_7 - x_8)|} + 1$$ there ...
0
votes
0answers
21 views

properties of vector space

Let $E$ be Banach space and $0<r<1$, $1\leq p<\infty$. Define the set $A$ as follow $$A=\left\{(x_j)_{j\in\mathbb{N}}\subset E :\sum\limits_{j=1}^{\infty}\left[\left|\left\langle ...
0
votes
5answers
72 views

Proving $x<y \implies n^{x}<n^{y}$, for n>1. $x,y \in \mathbb R$

I think I'm supposed to use the lowest upper bound property but I don't even know where to construct a set to start the problem.